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\section{\usemenu{slac-pub-7113::context::slac-pub-7113-0-0-1}{Introduction}}\label{section::slac-pub-7113-0-0-1}
\label{sec:introd}
In the Standard model (SM) flavor-changing neutral currents
only arise at the one-loop level. The corresponding B-meson decays
are therefore particularly
sensitive to 'new physics'; but also
within the Standard model framework, they can
be used to constrain several Cabibbo-Kobayashi-Maskawa matrix
elements involving the top-quark. For both reasons
precise experimental and theoretical work is of great importance.
$B \to K^* \gamma$ is the first rare B decay mode, which
has been measured in 1993 by the CLEO collaboration \cite{1}
and recently also the first measurement of
the inclusive photon energy
spectrum and the branching ratio
in the decay $\BGAMAXS$ was reported \cite{2}.
In contrast to the exclusive channels, the inclusive mode allows a less
model-dependent comparison with theory, because no specific
bound state model is needed for the final state.
This data is in a good
agreement with
the SM-based theoretical computations presented
in \cite{3,4,5}
given that large uncertainties exist in both
experimental and theoretical results.
In particular, the measured branching ratio $BR(B \to X_s \gamma)
= (2.32 \pm 0.67)
\times 10^{-4}$ \cite{2} overlaps with the
SM-based estimates in
\cite{3,4} and in \cite{6,7}.
As the experiments are becoming more precise in the near future,
also the calculations must be refined in order to
allow quantitative statements about new
physics or standard model
parameters.
It is well known that
QCD corrections to the decay rate for $b \to s \g$
bring in
large logarithms of the form $\a_s^n(m_W) \, \log^m(m_b/M)$,
where $M=m_t$ or $m_W$ and $m \le n$ (with $n=0,1,2,...$).
These large terms can be resummed by renormalization
group techniques.
At present, only the leading logarithmic
corrections (i.e. $m=n$) have been calculated systematically.
In this work we include one class of next-to-leading
corrections which we describe in more detail below.
The calculations are most easily done in the
framework of an effective theory which is obtained
by integrating out the top quark and the $W$-boson
in the standard model and other heavy particles
in extensions thereof.
A complete set of dimension-6 operators relevant for the process
$b \to s \gamma$ (and $b \to s \g g$)
is contained in the effective Hamiltonian
\cite{8}
\begin{equation}
\label{heff}
H_{eff}(b \to s \gamma)
= - \frac{4 G_{F}}{\sqrt{2}} \, \lambda_{t} \, \sum_{j=1}^{8}
C_{j}(\mu) \, O_j(\mu) \quad ,
\end{equation}
where
$G_F$ is the Fermi constant
coupling constant and
$C_{j}(\mu) $ are the Wilson coefficients evaluated at the scale $\mu$,
and $\lambda_t=V_{tb}V_{ts}^*$ with $V_{ij}$ being the
CKM matrix elements.
The operators $O_j$ read
\bea
\label{operators}
O_1 &=& \left( \bar{c}_{L \b} \g^\m b_{L \a} \right) \,
\left( \bar{s}_{L \a} \g_\m c_{L \b} \right)\,, \nonumber \\
O_2 &=& \left( \bar{c}_{L \a} \g^\m b_{L \a} \right) \,
\left( \bar{s}_{L \b} \g_\m c_{L \b} \right) \,,\nonumber \\
O_7 &=& (e/16\p^{2}) \, \bar{s}_{\a} \, \sigma^{\m \n}
\, (m_{b}(\mu) R + m_{s}(\mu) L) \, b_{\a} \ F_{\m \n} \,,
\nonumber \\
O_8 &=& (g_s/16\p^{2}) \, \bar{s}_{\a} \, \sigma^{\m \n}
\, (m_{b}(\mu) R + m_{s}(\mu) L) \, (\l^A_{\a \b}/2) \,b_{\b}
\ G^A_{\m \n} \quad ,
\nonumber \\
\eea
where $e$ and $g_s$ are the electromagnetic and the strong
coupling constants, respectively. In the magnetic moment type
operators $O_7$ and $O_8$, $F_{\m \n}$ and $G^A_{\m \n}$
are the electromagnetic and the gluonic field strength
tensors, respectively and
$L=(1-\g_5)/2$ and $R=(1+\g_5)/2$
stand for the left and right-handed projection operators.
We note here that the explicit mass factors in $O_7$
and $O_8$ are the running quark masses.
We
did not give explicitly
the four-Fermi operators $O_3$--$O_6$ in
eq. (\docLink{slac-pub-7113-0-0-1.tcx}[operators]{2}), because they have small
and negligible Wilson coefficients.
To leading logarithmic precision,
it is consistent to perform the
matching of the effective and full theory
without taking into account QCD-corrections
\cite{9}
and to calculate
the anomalous dimension matrix (8 $\times$ 8)
to order $\a_s$ \cite{10}.
The corresponding leading logarithmic Wilson coefficients
are given explicitly in
\cite{6,11}.
The leading logarithmic contribution to the
decay matrix element is then obtained by calculating the
tree-level matrix element of the operator $C_7 O_7$ and
the one-loop matrix elements of the four-Fermi operators
$C_i O_i$ ($i=1,...,6$).
In the NDR scheme which we will use in this paper,
the latter are non-zero only for $i=5,6$. Their effect
can be absorbed into a redefinition of $C_7 \to C_7^{eff}$
\footnote{For the analogous $b \to s g$ transition, the effects
of the four-Fermi operators can be absorbed by the shift
$C_8 \to C_8^{eff}=C_8 + C_5$.}
\be
\label{C78eff}
C_7^{eff} \equiv C_7 + Q_d \, C_5 + 3 Q_d \, C_6 \quad .
\ee
Since the first order calculations contain large
scale uncertainties,
it is important to take into account the next-to-leading
order corrections.
A complete next-to-leading calculation contains
two classes of improvements: First,
the Wilson coefficients are required to next-leading
order at the scale $\mu \approx m_b$. This requires
the matching
with the full theory (at $\mu=m_W$)
at the $O(\a_s)$ level and the renormalization
group equation has to be solved using the anomalous dimension
matrix calculated up to order $\a_s^2$.
Second,
the real and virtual $O(\a_s)$ corrections for the matrix element (at scale
$\mu \approx m_b$)
must be evaluated.
The higher order matching has been calculated in ref. \cite{12}
and work on the Wilson coefficients is in progress.
In the present paper we complete
the second step.
While the Bremsstrahlung corrections have been worked out
\cite{3,4,13,14}
in order to
get a non-trivial photon energy spectrum at the partonic level
for $B \to X_s \g$, the virtual corrections to $b \to s \g$
have not been completely known so far. Only those
virtual diagrams which are needed to
cancel the infrared sigularities generated by the Bremsstrahlung
diagram were calculated.
In the present paper we
evaluate all the additional virtual
correction, neglecting, as mentioned the contributions of the small
operators
\footnote{
This omission will be a source of a slight scheme and scale dependence
of the next-to-leading order result.} $O_3$--$O_6$.
These new contributions
substantially
reduce the strong scale dependence of the leading calculation.
In the following we thus consider the virtual $O(\a_s)$
corrections to $b \to s \g$ due to
the four-Fermi operator $O_2$
and the
magnetic operators $O_7$ (which has already been calculated in the
literature) and $O_8$ (which is new); note
that the operator $O_1$ does not contribute
to the matrix elements for $b \to s \g$ and $b \to s \g g$
because of its color structure.
As the corrections to $O_7$ and $O_8$ are one-loop diagrams,
they are relatively easy to work out.
In contrast, the corrections to $O_2$, involve two-loop
diagrams.
% since this operator itself only contributes
%at the one-loop level.
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